Question

6. A singular point z=z0 is said to an ---- singular point of f(z) if lim f(z) exists
and finite
a)poles
b) isolated
c)Removable
d)none
7. A singular point z=20
is said to be an- singular point of f(z) it is neither an
isolated singularity nor a removable singularity
a)poles
b) isolated
c)Removable
d) essential
8. If f(a)=0 and f(a) #0 the z-a is called a
a) simple zero
b) simple curve
c) zero of order n
d) none​

Answer 1

Answer:

6 ans Removable

7 ans. Essential

8 ans. zero of ordern

Answer 2

Answer:

6. Option (c)

7. Option (d)

8. Option (c)

Step-by-step explanation:

Recall the definition of types of singularities,

Isolated singularity: $f(z)$ has isolated singularity at $z=z_0$ if $f(z)$ is not analytic at $z_0$ but it is analytic throughout some deleted neighborhood of $z_0$.

Removable singularity: If $f(z)$ has an isolated singularity at $z_0$ then the point  $z=z_0$ is a removable singularity if

$\lim_{z \to z_{0}} (z-z_0)f(z)$  exists and is finite.

Pole: If $z=z_0$ is an isolated singularity of $f$ then $z_0$ is a pole of $f$ if

$\lim_{z \to z_{0}}|f(z)|=\infty$.

Essential singularity: If an isolated singularity is neither a pole nor a removable singularity, it is called an essential singularity.

6. From the above definitions,

A singular point $z=z_0$ is said to a removable singular point of $f(z)$ if $\lim_{z \to z_{0}} f(z)$ exists and finite.

Option (c) is correct.

7. From the above definitions,

A singular point $z=z_0$ is said to be an essential singular point of $f(z)$ if it is neither an isolated singularity nor a removable singularity.

Option (d) is correct.

8.

(a) Simple zero or simple pole are terms used for zeroes and poles of order.

(b) A simple curve is a curve that does not cross itself.

(c) A point $z_0$ is called a pole of order $n$ of $f(z)$ if $\frac{1}{f}$ has a zero of order $n$ at $z_0$.

Therefore, option (c) is correct.

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